A358727 - OEIS

%I #5 Dec 02 2022 07:06:09

%S 8,16,24,28,32,36,38,42,48,49,53,54,56,57,63,64,72,76,80,81,84,96,98,

%T 104,106,108,112,114,120,126,128,131,133,136,140,144,147,148,152,156,

%U 159,160,162,168,171,172,178,180,182,184,189,190,192,196,200,204,208

%N Matula-Goebel numbers of rooted trees with greater number of leaves (width) than node-height.

%C The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

%C Node-height is the number of nodes in the longest path from root to leaf.

%H Gus Wiseman, <a href="/A358727/a358727.png">The first 64 rooted trees with greater width than height.</a>

%e The terms together with their corresponding rooted trees begin:

%e    8: (ooo)

%e   16: (oooo)

%e   24: (ooo(o))

%e   28: (oo(oo))

%e   32: (ooooo)

%e   36: (oo(o)(o))

%e   38: (o(ooo))

%e   42: (o(o)(oo))

%e   48: (oooo(o))

%e   49: ((oo)(oo))

%e   53: ((oooo))

%e   54: (o(o)(o)(o))

%e   56: (ooo(oo))

%e   57: ((o)(ooo))

%e   63: ((o)(o)(oo))

%e   64: (oooooo)

%e   72: (ooo(o)(o))

%e   76: (oo(ooo))

%t MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[1000],Depth[MGTree[#]]-1<Count[MGTree[#],{},{-2}]&]

%Y Positions of negative terms in A358726.

%Y These trees are counted by A358728.

%Y Differences: A358580, A358724, A358726, A358729.

%Y A000081 counts rooted trees, ordered A000108.

%Y A034781 counts rooted trees by nodes and height, ordered A080936.

%Y A055277 counts rooted trees by nodes and leaves, ordered A001263.

%Y MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.

%Y MG core: A000040, A000720, A001222, A007097, A056239, A112798.

%Y Cf. A185650, A206487, A209638, A358576, A358577, A358578, A358587, A358730.

%K nonn

%O 1,1

%A _Gus Wiseman_, Dec 01 2022